1877.] electricity through gases. " 61 



parts of the stratification therefore would account for the different 

 spectra. I am unable to say why a difference of temperature 

 should produce the change, but there is as yet no reason to 

 suppose that other causes but those of temperature determine the 

 different spectra in wide and narrow parts of the tube. It would 

 therefore seem that the temperature of that part of the stratifica- 

 tion which is turned towards the positive metallic electrode is at a 

 higher temperature than that which is turned away from it, and 

 this agrees with the observations of Mr Goldstein, who found that 

 the stratifications were always brightest at their negative pole, 

 that is, the side which is turned towards the metallic positive 

 electrode. 



May 7, 1877. 



Prof, Clerk Maxwell, F.E.S., President, in the chair. 



A communication was made to the Society by 



Mr J. W. L. Glaisher, M.A., F.R.S. On expressions for the 

 theta functions as definite integrals. 



(Abstract) 

 § 1. The function (x) is defined by the equation 



j = l-2q COS 2x + 2^* cos 4!X - 2q' cos 6x + &c. ...(1), 



so that the object of the memoir is to express the series on the 

 right-hand side of (1) as a definite integral. 



The subject is alluded to by Kummer in his paper, De inte- 

 gralibus definitis et seriebus injinitis (Crelle's Journal, t. xvii. pp. 

 210 — 242). He there gives the values of the series 



l+q + q' + q' + q'' + &c. 



and l-q + q*-q^ + q'^-&c. 



as definite integrals, and remarks that the same method would 



enable us to find the values of the series for © ( j and 



H ( j, 'quarum vero expressiones per integralia definita, 



quam minus simplices evadant, hoc loco omittimus.' 



